To calculate the solubility product constant (Ksp) of the salt \( AB_2 \), we can follow these steps:
### Step 1: Determine the dissociation of the salt
The salt \( AB_2 \) dissociates in water according to the following equation:
\[
AB_2 (s) \rightleftharpoons A^{2+} (aq) + 2B^{-} (aq)
\]
### Step 2: Write the expression for Ksp
The solubility product constant \( K_{sp} \) can be expressed as:
\[
K_{sp} = [A^{2+}][B^{-}]^2
\]
### Step 3: Define the solubility
Let the solubility of \( AB_2 \) be \( s \) (in moles per liter). From the dissociation equation, we can express the concentrations at equilibrium:
- The concentration of \( A^{2+} \) will be \( s \).
- The concentration of \( B^{-} \) will be \( 2s \) (since 2 moles of \( B^{-} \) are produced for every mole of \( AB_2 \)).
### Step 4: Substitute into the Ksp expression
Substituting the concentrations into the Ksp expression gives:
\[
K_{sp} = [s][2s]^2 = s(4s^2) = 4s^3
\]
### Step 5: Calculate the molar mass of \( AB_2 \)
The molar mass of \( AB_2 \) can be calculated as follows:
- Atomic mass of \( A = 20 \, \text{g/mol} \)
- Atomic mass of \( B = 40 \, \text{g/mol} \)
Thus, the molar mass of \( AB_2 \) is:
\[
\text{Molar mass of } AB_2 = 20 + 2 \times 40 = 20 + 80 = 100 \, \text{g/mol}
\]
### Step 6: Convert solubility from grams to moles
Given the solubility of \( AB_2 \) is \( 0.05 \, \text{g} \) per \( 100 \, \text{mL} \), we convert this to moles:
\[
\text{Solubility in moles} = \frac{0.05 \, \text{g}}{100 \, \text{mL}} \times \frac{1000 \, \text{mL}}{1 \, \text{L}} \times \frac{1 \, \text{mol}}{100 \, \text{g}} = \frac{0.05}{100} \times 10 = 5 \times 10^{-3} \, \text{mol/L}
\]
Thus, \( s = 5 \times 10^{-3} \, \text{mol/L} \).
### Step 7: Calculate Ksp
Now we can substitute \( s \) back into the Ksp expression:
\[
K_{sp} = 4s^3 = 4(5 \times 10^{-3})^3
\]
Calculating \( (5 \times 10^{-3})^3 \):
\[
(5 \times 10^{-3})^3 = 125 \times 10^{-9} = 1.25 \times 10^{-7}
\]
Thus,
\[
K_{sp} = 4 \times 1.25 \times 10^{-7} = 5 \times 10^{-7}
\]
### Final Answer:
\[
K_{sp} = 5 \times 10^{-7}
\]
